For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer ${\displaystyle f^{*}}$ of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data.

The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola:

Theorem: Consider a positive-definite real-valued kernel ${\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} }$ on a non-empty set ${\displaystyle {\mathcal {X}}}$ with a corresponding reproducing kernel Hilbert space ${\displaystyle H_{k}}$. Let there be given

which together define the following regularized empirical risk functional on ${\displaystyle H_{k}}$:

Then, any minimizer of the empirical risk

admits a representation of the form:

where ${\displaystyle \alpha _{i}\in \mathbb {R} }$ for all ${\displaystyle 1\leq i\leq n}$.

Proof: Define a mapping